Control Polygon Research Articles

Curvature continuous corner cutting

Subdivision schemes are used to generate smooth curves by iteratively refining an initial control polygon. The simplest such schemes are corner cutting schemes, which specify two distinct points on each edge of the current polygon and connect them to get the refined polygon, thus cutting off the corners of the current polygon. While de Boor (1987) shows that this process always converges to a Lipschitz continuous limit curve, no matter how the points on each edge are chosen, Gregory and Qu (1996) discover that the limit curve is continuously differentiable under certain constraints. We extend these results and show that the limit curve can even be curvature continuous for specific sequences of cut ratios.

Continuous Scatterplot Operators for Bivariate Analysis and Study of Electronic Transitions.

Electronic transitions in molecules due to the absorption or emission of light is a complex quantum mechanical process. Their study plays an important role in the design of novel materials. A common yet challenging task in the study is to determine the nature of electronic transitions, namely which subgroups of the molecule are involved in the transition by donating or accepting electrons, followed by an investigation of the variation in the donor-acceptor behavior for different transitions or conformations of the molecules. In this article, we present a novel approach for the analysis of a bivariate field and show its applicability to the study of electronic transitions. This approach is based on two novel operators, the continuous scatterplot (CSP) lens operator and the CSP peel operator, that enable effective visual analysis of bivariate fields. Both operators can be applied independently or together to facilitate analysis. The operators motivate the design of control polygon inputs to extract fiber surfaces of interest in the spatial domain. The CSPs are annotated with a quantitative measure to further support the visual analysis. We study different molecular systems and demonstrate how the CSP peel and CSP lens operators help identify and study donor and acceptor characteristics in molecular systems.

Geometric Modelling of a Family of 3-point Quaternary Subdivision Schemes Rζ

Computer-aided geometric design combines mathematical concepts and computing skills that smooth curves through subdivision schemes. Subdivision schemes perform smoothing by turning the control polygon into a limit curve under a refinement rule, a prime example of which is improving the signal-to-noise ratio in modern devices. Because of the importance and location of subdivision schemes, mathematicians use them in CAD, computer graphics and advanced simulation methods. In this research, a family of 3-point Quaternary approximating subdivision scheme $R_{\zeta}$ is presented with its properties and analysis, including necessary conditions for convergence, Laurent polynomial, degree of the generation and polynomial reproduction, continuity analysis, Hölder regularity, and limit stencils. The visual performance of the proposed scheme is also presented to highlight the importance of this research and to validate the scheme.

Euler Bézier spirals and Euler B-spline spirals

Euler spirals have linear varying curvature with respect to arc length and can be applied in fields such as aesthetics pleasing shape design, curve completion or highway design, etc. However, evaluation and interpolation of Euler spirals to prescribed boundary data is not convenient since Euler spirals are represented by Fresnel integrals but with no closed-form expression of the integrals. We investigate a class of Bézier or B-spline curves called Euler Bézier spirals or Euler B-spline spirals which have specially defined control polygons and approximate linearly varying curvature. This type of spirals can be designed conveniently and evaluated exactly. Simple but efficient algorithms are also given to interpolate G1 boundary data by Euler Bézier spirals or cubic Euler B-spline spirals.

Subdivision algorithms with modular arithmetic

We study the de Casteljau subdivision algorithm for Bezier curves and the Lane-Riesenfeld algorithm for uniform B-spline curves over the integers mod m, where m>2 is an odd integer. We place the integers mod m evenly spaced around a unit circle so that the integer k mod m is located at the position on the unit circle ate2πki/m=cos(2kπ/m)+isin(2kπ/m)↔(cos(2kπ/m),sin(2kπ/m)).Given a sequence of integers (s0,…,sm)mod m, we connect consecutive values sjsj+1 on the unit circle with straight line segments to form a control polygon. We show that if we start these subdivision procedures with the sequence (0,1,…,m)mod m, then the sequences generated by these recursive subdivision algorithms spawn control polygons consisting of the regular m-sided polygon and regular m-pointed stars that repeat with a period equal to the minimal integer k such that 2k=±1modm. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue 2−1modm. We go on to study the effects of these subdivision procedures on more general initial control polygons, and we show in particular that certain control polygons, including the orbits of regular m-sided polygons and the complete graphs of m-sided polygons, are fixed points of these subdivision procedures.

ЖЕРДІ ҚАШЫҚТЫҚТАН ЗОНДТАУ ӘДІСТЕРІН ПАЙДАЛАНА ОТЫРЫП, АУЫЛ ШАРУАШЫЛЫҒЫ АЛҚАПТАРЫНАН ДИФФУЗИЯЛЫҚ ЖЕР ҮСТІ АҒЫНЫНЫҢ ҚАРАТОМАР СУ ҚОЙМАСЫ СУЫНЫҢ САПАСЫНА ӘСЕРІН МОНИТОРИНГІЛЕУДІҢ НЕГІЗГІ ТӘСІЛДЕРІ

The article describes the implementation of a new methodological approach for assessing diffuse runoff of biogenic elements using calculated NDMI and NDCI L1 indices derived from satellite imagery data. The study focuses on analyzing the inflow of biogenic elements into the Karatomar Reservoir from agricultural fields by identifying control polygons in raster images and indexing them based on NDMI on land and NDCI L1 in the reservoir's water area. This method allowed the tracking of the formation and inflow of biogenic elements with surface runoff during the period from May to October 2021-2023, identifying the primary sources of pollution. The calculation of correlation significance indices showed high accuracy in the data obtained when analyzing the relationship between NDMI and NDCI L1 indices. During the research, regulatory requirements and scientific recommendations for data collection and processing using remote sensing methods were studied. This methodology could later serve as the basis for developing an automated system to monitor the diffuse flow of biogenic elements of anthropogenic or technogenic origin, using remote sensing tools. The goal would be to create mechanisms to prevent or reduce pollution of various surface water bodies, including lotic and lentic ecosystems.

New generalized blended trigonometric Bézier curves with one shape parameter

This paper introduces a new basis for generalized blended trigonometric (GBT) B?zier curves along with one shape parameter. The recursive technique is adopted to formulate the basis for higher order GBT-B?zier curves. The curves are better approximated using the proposed basis than the traditional Bernstein basis. New basis functions and curves satisfy all the properties followed by classical B?zier curves. The shape of these curves can be adjusted by changing the values of the parameter, keeping the control polygon unchanged. This adds to the flexibility of new GBT-B?zier curves. Appropriate conditions for parametric (C0,C1,C2 and C3) and geometric (G0, G1, G2 and G3) continuities to compose two or more GBT-B?zier curves have been worked upon. Applications of the proposed GBT-B?zier curves are discussed with different formations.

Conics in rational cubic Bézier form made simple

We revisit the rational cubic Bézier representation of conics, simplifying and expanding previous works, elucidating their connection, and making them more accessible. The key ingredient is the concept of conic associated with a given (planar) cubic Bézier polygon, resulting from an intuitive geometric construction: Take a cubic semicircle, whose control polygon forms a square, and apply the perspective that maps this square to the given polygon. Since cubic conics come from a quadratic version by inserting a base point, this conic admitting the polygon turns out to be unique. Therefore, detecting whether a cubic is a conic boils down to checking out whether it coincides with the conic associated with its control polygon. These two curves coincide if they have the same shape factors (aka, shape invariants) or, equivalently, the same oriented curvatures at the endpoints. Our results hold for any cubic polygon (with no three points collinear), irrespective of its convexity. However, only polygons forming a strictly convex quadrilateral define conics whose cubic form admits positive weights. Also, we provide a geometric interpretation for the added expressive power (over quadratics) that such cubics with positive weights offer. In addition to semiellipses, they encompass elliptical segments with rho-values over the negative unit interval.

New algebraic and geometric characterizations of planar quintic Pythagorean-hodograph curves

The aim of this work is to provide new characterizations of planar quintic Pythagorean-hodograph curves. The first two are algebraic and consist of two and three equations, respectively, in terms of the edges of the Bézier control polygon as complex numbers. These equations are symmetric with respect to the edge indices and cover curves with generic as well as degenerate control polygons. The last two characterizations are geometric and rely both on just two auxiliary points outside the control polygon. One requires two (possibly degenerate) quadrilaterals to be similar, and the other highlights two families of three similar triangles. All characterizations are a step forward with respect to the state of the art, and they can be linked to the well-established counterparts for planar cubic Pythagorean-hodograph curves. The key ingredient for proving the aforementioned results is a novel general expression for the hodograph of the curve.

Curve fitting by GLSPIA

We develop the generalized B-splines least square progressive iterative approximation (GLSPIA) method to improve LSPIA method (Deng and Lin, 2014) from the perspective of its equivalent neural network framework. We replace the B-spline basis function in standard LSPIA method with the generalized B-spline basis function (Juhász and Róth, 2013) in the GLSPIA method. By adjusting the control points as well as the parameter in the core function, higher precision fitting is achieved. Moreover, in order to solve the over-fitting problem of GLSPIA method when dealing with noise data, we propose three kinds of regularizations using the geometric properties of control polygon. Numerical examples are presented to show the effectiveness of our method.

The Quadratic Trigonometric Bezier like Curve With two Shape Parameters

The quadratic trigonometric Bezier model serves as a potent and valuable tool in computer-aided geometric design, boasting exceptional geometric properties. In this study, Novel functions are introduced by leveraging a trigonometric Bezier-like curve with adjustable shape parameters. Each curve segment is formed by three consecutive control points, providing precise control over shape by adjusting these parameters while keeping the control polygon unchanged. These curves exhibit superior proximity to the control polygon compared to quadratic Bezier curves across all shape parameter values, gradually approaching the control polygon as the shape parameter increases. Moreover, for practical application, the quadratic trigonometric Bezier-like curve was utilized to artfully design the Arabic word "الله" (Allah) and define flower contours, employing MATHEMATICA software. The results underscore the model's effectiveness in accurately representing intricate shapes, offering a versatile and efficient solution for diverse design challenges. Furthermore, the fusion of the quadratic trigonometric Bezier-like curve with shape parameters significantly augments design flexibility and adaptability, making it a valuable asset in computer graphics, geometric modeling, and artistic design across multiple domains.

B/Surf: Interactive Bézier Splines on Surface Meshes.

We present a practical framework to port Bzier curves to surfaces. We support the interactive drawing and editing of Bzier splines on manifold meshes with millions of triangles, by relying on just repeated manifold averages. We show that direct extensions of the de Casteljau and Bernstein evaluation algorithms to the manifold setting are fragile, and prone to discontinuities when control polygons become large. Conversely, approaches based on subdivision are robust and can be implemented efficiently. We implement manifold extensions of the recursive de Casteljau bisection, and an open-uniform Lane-Riesenfeld subdivision scheme. For both schemes, we present algorithms for curve tracing, point evaluation, and approximated point insertion. We run bulk experiments to test our algorithms for robustness and performance, and we compare them with other methods at the state of the art, always achieving correct results and superior performance. For interactive editing, we port all the basic user interface interactions found in 2D tools directly to the mesh. We also support mapping complex SVG drawings to the mesh and their interactive editing.

Construction of planar quintic Pythagorean-hodograph curves by control-polygon constraints

In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈[0,1] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves.

Shape control tools for periodic Bézier curves

Bézier curves are an essential tool for curve design. Due to their properties, common operations such as translation, rotation, or scaling can be applied to the curve by simply modifying the control polygon of the curve. More flexibility, and thus more diverse types of curves, can be achieved by associating a weight with each control point, that is, by considering rational Bézier curves. As shown by Ramanantoanina and Hormann (2021), additional and more direct control over the curve shape can be achieved by exploiting the correspondence between the rational Bézier and the interpolating barycentric form and by exploring the effect of changing the degrees of freedom of the latter (interpolation points, weights, and nodes). In this paper, we explore similar editing possibilities for closed curves, in particular for the rational extension of the periodic Bézier curves that were introduced by Sánchez-Reyes (2009). We show how to convert back and forth between the periodic rational Bézier and the interpolating trigonometric barycentric form, derive a necessary condition to avoid poles of a trigonometric rational interpolant, and devise a general framework to perform degree elevation of periodic rational Bézier curves. We further discuss the editing possibilities given by the trigonometric barycentric form.

G1 interpolation of v-asymmetric data with arc-length constraints by Pythagorean-hodograph cubic splines

The interpolation problem of a sequence of points with prescribed segment arc-length by PH cubic spline is addressed. It turns out to be a v-asymmetric G1 Hermite interpolation problem with specified arc length. Using rectifying control polygon and reduction to a canonical form, it is shown that the problem can be expressed in terms of finding the real solutions to a system of nonlinear equations in two variables. A detailed and thorough analysis of the resulting system of nonlinear equations and closed form solutions are provided for any given data. It is confirmed that this construction of G1 Hermite interpolants of specified arc length admits at least two formal solutions, both of which have attractive shape properties and some (if any) must be discarded due to undesired looping behavior. The algorithm developed herein offers simple and efficient closed-form solutions to a fundamental constructive geometry problem that avoids the need for iterative numerical methods. Moreover, the algorithm is illustrated by several numerical examples.

3D Class A Bézier curves with monotone curvature

This paper introduces a special class of 3D Bézier curves that are defined by their degree, a starting point, the first leg of their control polygons, and a 3D affine transformation composing of a uniform scaling and a rotation. We present new formulas for the curvature of such Bézier curves and based on the new formulas we derive sufficient conditions for the curves to have monotonic curvature. The conditions are expressed by a simple constraint on the rotation angle and the scaling factor. This facilitates constructing 3D Class A Bézier curves that are a generalization of planar typical curves proposed by Mineur et al. (1998), which are often used in automotive and other design applications. Some examples are provided to demonstrate the effectiveness of our construction.

Polynomial-Based Non-Uniform Ternary Interpolation Surface Subdivision on Quadrilateral Mesh

For non-uniform control polygons, a parameterized four-point interpolation curve ternary subdivision scheme is proposed, and its convergence and continuity are demonstrated. Following curve subdivision, a non-uniform interpolation surface ternary subdivision on regular quadrilateral meshes is proposed by applying the tensor product method. Analyses were conducted on the updating rules of parameters, proving that the limit surface is continuous. In this paper, we present a novel interpolation subdivision method to generate new virtual edge points and new face points of the extraordinary points of quadrilateral mesh. We also provide numerical examples to assess the validity of various interpolation methods.

On Control Polygons of Planar Sextic Pythagorean Hodograph Curves

In this paper, we analyze planar parametric sextic curves to determine conditions for Pythagorean hodograph (PH) curves. By expressing the curves to be analyzed in the complex form, the analysis is conducted in algebraic form. Since sextic PH curves can be classified into two classes according to the degrees of their derivatives’ factors, we introduce auxiliary control points to reconstruct the internal algebraic structure for both classes. We prove that a sextic curve is completely characterized by the lengths of legs and angles formed by the legs of their Bézier control polygons. As such conditions are invariant under rotations and translations, we call them the geometric characteristics of sextic PH curves. We demonstrate that the geometric characteristics form the basis for an easy and intuitive method for identifying sextic PH curves. Benefiting from our results, the computations of the parameters of cusps and/or inflection points can also be simplified.

Non-uniform subdivision schemes of ωB-spline curves and surfaces with variable parameters

In this paper, we propose a new non-uniform subdivision scheme that includes a tension parameters sequence. Each tension parameter of the sequence is assigned at each edge of the initial control polygon. The proposed scheme can produce limiting curves that are more consistent with the original data points and the control polygon. It has also the advantage of generating a wide variety of shapes for the limiting curves. Moreover, to produce smooth limit surfaces, the proposed scheme is firstly extended to tensor-product surfaces, then to meshes of arbitrary topology. The convergence and smoothness of the proposed scheme are proven. Numerical results that illustrate the advantages of the proposed non-uniform scheme are given. • This family of non-uniform subdivision schemes accompanied with algorithms for curves and surfaces can be used in many engineering applications. • It is able to generate a wide variety of shapes for the limiting curves and surfaces as it includes a tension parameters sequence. • It allows local control of the shape of the resulting limit curves and surfaces. • It is extended to meshes of arbitrary topology to generate complex surface subdivisions. • It turns out to be the uniform scheme, when the considered tension parameters in the sequence are all equals. • It turns out to be several well-known subdivision schemes, such as Doo–Sabin and Catmull–Clark subdivision, when all the tension parameters in the sequence are equals to 1.

Characteristics of planar sextic indirect-PH curves

<abstract><p>By a fractional quadratic transformation, an indirect-PH curve can have rational offsets. In this paper, I study properties of planar sextic indirect-PH curves, in terms of their Bézier control polygon legs. With our results, sextic Bézier curves can be efficiently tested whether they are indirect-PH curves. The main strategy to achieve our results is using complex representation of planar parametric curve. Sextic indirect-PH curves can be classified into three classes according to different factorizations of their hodographs. Necessary and sufficient conditions for all classes of sextic indirect-PH curves can be described by non-linear complex systems. By analyzing these non-linear systems, algebraic conditions for a sextic Bézier curve to be an indirect-PH curve are first discussed, then geometric characteristics in terms of legs of its control polygon are revealed.</p></abstract>